When does the universal-transfer reduction hold for categorical transfer systems?

Determine necessary and sufficient conditions under which the analogue of the universal‑transfer reduction (Corollary reducing the computation of M(O) to the interval [N_O, G] followed by restriction) holds for categorical transfer systems on a bounded lattice, thereby characterizing precisely when such a reduction applies in the categorical (non‑G) setting.

Background

For G‑transfer systems, the authors prove that computing the maximal compatible transfer system M(O) for a disklike O can be reduced to a smaller interval [N_O, G] via inflation and restriction (Corollary 5.14).

They exhibit a categorical (non‑G) counterexample where the naive analogue fails and explicitly note that understanding when an analogue does hold is an open question.